Frequentist Spatial Scan Statistic

• The framework assumes that we observe counts \(z_i\) such that \(z_i \sim \text{Poisson}(qb_i)\)
  • Where \(b_i\) represents the known baseline/at risk population of location \(i\)
  • \(q\) is the unknown underlying disease rate

\[ H_0: \text{No cluster (common rate for all regions)} \\ H_1(S): \text{Cluster in subset }S\text{ with elevated rate vs. outside } S \]

• Compute likelihood ratio test statistic for each candidate zone \(S\)
• The scan statistic test statistic is \(\Lambda = \max_{S \in C}\lambda(S)\).
• Generate Monte Carlo samples under \(H_0\) to calculate P-value

Bayesian Spatial Scan Statistics

• Assuming we observe count data \(z_i\) in area \(i\), each associated with baseline \(b_i\)
• Under the null hypothesis there is no cluster and all locations share \(q_{all}\) \[ z_i \sim \text{Poisson}(q_{all} b_i), \quad q_{all} \sim \text{gamma}(\alpha_{all}, \beta_{all})\]
• The alternative hypothesis for each candidate cluster \(S \in \mathcal{S}\), where \(\mathcal{S}\) is the space of all possible clusters \[ \begin{cases} z_i \sim \text{Poisson}(q_{in} b_i), &i \in S, \quad q_{in} \sim \text{Gamma}(\alpha_{in}, \beta_{in}), \\ z_i \sim \text{Poisson}(q_{out} b_i), &i \notin S, \quad q_{out} \sim \text{Gamma}(\alpha_{out}, \beta_{out}). \end{cases} \]
• Marginal likelihoods based on the gamma-Poisson model
• Conjugate model possible to solve for closed form solution

Bayesian Spatial Scan Statitic Testing

• Using the maringal likelihoods from the models the posterior probability under the null is \[P(H_0 | D) = \frac{P(D|H_0) P(H_0)}{P(D)}\]
• The posterior probability under the alternative is \[P(H_1(S) | D) = \frac{P(D|H_1(S)) P(H_1(S))}{P(D)}\]
• Then we can return regions with non-negligible posterior probabilities
• Since we have the full posterior probability distributions there is no need for randomization testing
• Bayes factors can be used to provide a direct measure of evidence for one hypothesis over the other

Bayesian Spatial Scan Statistic Extension

• The new null hypotheses assumes no clusters \[ z_i \sim \text{Poisson}(p \times q_{all} \times b_i), \quad q_{all} \sim \text{gamma}(\alpha_{all}, \beta_{all}), \quad p \sim \text{beta}(\alpha, \beta) \]
• The resulting alternative hypotheses for candidate region \(S\) is \[ \begin{cases} z_i \sim \text{Poisson}(p \times q_{in} \times b_i), &i \in S, \quad q_{in} \sim \text{Gamma}(\alpha_{in}, \beta_{in}), \\ z_i \sim \text{Poisson}(p \times q_{out} \times b_i), &i \notin S, \quad q_{out} \sim \text{Gamma}(\alpha_{out}, \beta_{out}). \end{cases} \\ p \sim \text{beta}(\alpha, \beta) \]

Posterior Estimation

• The marginal likelihood of the null hypothesis is now: \[ P(D|H_0) = \int \int \pi(q_{all}) \times \pi(p) \times \prod_{i \in \mathcal{S}} P(D|q_{all}, p, b_i) dq_{all} dp \]
• The marginal likelihood under a candidate region \(S\) is now: \[ P(D|H_1(S)) = \int \int \pi(q_{in}) \times \pi(p) \times \prod_{i \in S} P(D|q_{in}, p, b_i) dq_{in} dp \times \\ \quad \int \int \pi(q_{out}) \times \pi(p) \times \prod_{i \in \mathcal{S} - S} P(D|q_{out}, p, b_i) dq_{out} dp \]
• Numerically approximated with MCMC sampling in stan

Simulation priors

• Priors for reporting rate \(p\) where set using the betabuster tool in epiR package \[ p \sim \text{Beta}(3.5, 23) \quad \text{if} \quad p = 0.1 \\ p \sim \text{Beta}(4.5, 15) \quad \text{if} \quad p = 0.2 \\ p \sim \text{Beta}(10, 22) \quad \text{if} \quad p = 0.3 \\ p \sim \text{Beta}(13, 19) \quad \text{if} \quad p = 0.4 \\ p \sim \text{Beta}(1, 1) \quad \text{if} \quad p = 0.5 \]
• Priors for \(q\) \[ q_{all} \sim \text{gamma}(2, 0.5) \\ q_{out} \sim \text{gamma}(2, 0.4) \\ q_{in} \sim \text{gamma}(2, 0.5) \]

Texas COVID-19 Data

• COVID-19 data in early 2020 were severely under-reported due to limited testing and difficulty to diagnose Hortaçsu, Liu, and Schwieg (2021)
• Data (254 Counties)
  • COVID-19 cases (Probable and Confirmed)
  • Population

Real Data (priors)

• Estimates from early COVID-19 studies suggest very low reporting rates (\(\approx 10\%\)), with low probability of exceeding 30\(\%\) Chen, Song, and Stamey (2022).
• This information results in a prior of \(p \sim \text{Beta}(3.5, 23)\)
• Diffuse priors where fit to \(q_\cdot\) parameters

\[ q_{all} \sim \text{gamma}(1.5, 1) \\ q_{out} \sim \text{gamma}(1.5, 1) \\ q_{in} \sim \text{gamma}(1.5, 1) \]

Real Data Results

Appendix: Equations

\[ P(D|H_0) \propto \frac{\beta_{all}^{\alpha_{all}}}{\Gamma(\alpha_{all}) \mathcal{B}(\alpha, \beta)} \int \int q_{all}^{\alpha_{all} + C - 1} p^{\alpha + C - 1} (1 - p)^{\beta - 1} e^{-q_{all}\beta_{all} - p q_{all} B} dp dq_{all} \] \[ P(D|H_1(S)) \propto \frac{\beta_{in}^{\alpha_{in}}}{\Gamma(\alpha_{in}) \mathcal{B}(\alpha, \beta)} \int \int q_{in}^{\alpha_{in} + C_S - 1} p^{\alpha + C_S - 1} (1 - p)^{\beta - 1} e^{-q_{in}\beta_{in} - p q_{in} B_S} dp dq_{in} \times \\ \quad \frac{\beta_{out}^{\alpha_{out}}}{\Gamma(\alpha_{out}) \mathcal{B}(\alpha, \beta)} \int \int q_{out}^{\alpha_{out} + C_{S-G} - 1} p^{\alpha + C_{S-G} - 1} (1 - p)^{\beta - 1} e^{-q_{out}\beta_{out} - p q_{out} B_{S-G}} dp dq_{out} \]

\[ P(D) = P(D|H_0) P(H_0) + \sum_{s_i \in G} P(D|H_1(S)) P(H_1(S)) \]